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Theorem smodm2 6160
Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smodm2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )

Proof of Theorem smodm2
StepHypRef Expression
1 smodm 6156 . 2  |-  ( Smo 
F  ->  Ord  dom  F
)
2 fndm 5192 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
3 ordeq 4264 . . . 4  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
42, 3syl 14 . . 3  |-  ( F  Fn  A  ->  ( Ord  dom  F  <->  Ord  A ) )
54biimpa 294 . 2  |-  ( ( F  Fn  A  /\  Ord  dom  F )  ->  Ord  A )
61, 5sylan2 284 1  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   Ord word 4254   dom cdm 4509    Fn wfn 5088   Smo wsmo 6150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-fn 5096  df-smo 6151
This theorem is referenced by: (None)
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